Abstract

We study the position and motion of the bow
shock during the passage of the 18-19 October 1995,
interplanetary magnetic cloud. The Geotail spacecraft made 26 bow
shock crossings while it was nominally crossing the dawnside
magnetosheath on a west-east trajectory approaching the
Sun-Earth line. Interplanetary parameters are measured by the
Wind spacecraft. The effects of changes in solar wind dynamic
pressure and magnetosonic Mach number, which fall into three
groups depending on interplanetary conditions, are studied and
their respective effects are separated. Observed bow shock
positions are compared with the model bow shock standoff
distances
[after
Cairns and Lyon, 1995]
and show good agreement.
Finally, we calculate the magnetopause standoff distance on the
basis of pressure balance between solar wind dynamic pressure and
the Earth magnetic field magnetic pressure and compare these
results with a magnetopause standoff distance derived from the
Shue et al. [1998]
model.
We find that the magnetopause standoff distance
during the cloud passage is larger than the
Shue et al. [1998]
result. We
attribute this to magnetosphere erosion and note that solar wind
conditions on this day show all prerequisites necessary
for erosion.

Magnetic Cloud Event, 18-19 October 1995

Figure 1

The magnetic cloud that passed Earth on 18-19 October 1995
(see
Figure 1),
caused the largest geomagnetic storm in the period between
1994 and 1997 ( Dst= -120 nT). The cloud
has been intensely studied
by the scientific community
[see, e.g.,
Burlaga et al., 1998;
Farrugia et al., 1998;
Lepping et al., 1997].
It was observed by the Wind spacecraft
upstream of Earth when the spacecraft was located at an
average radial distance of ~175
RE.

Magnetic clouds are very useful for investigating the interaction
between the solar wind and the magnetosphere because of their
special properties, which allow them to couple energy and
momentum to the magnetosphere, thus driving storms and substorms.
Interplanetary magnetic clouds are
characterized by
(1) strong magnetic field strengths relative to ambient values,
(2) low proton
b and proton temperature, and
(3) large and smooth rotation of magnetic field direction
[Burlaga et al., 1981;
Lepping et al.,1990].

Their passage at Earth typically lasts about 1-2 days,
and their dimension
at AU is ~0.25 AU.
Furthermore, magnetic clouds are often a dramatic source of long-lasting,
strong, negative
Bz of interplanetary magnetic field, which is an
optimum condition for reconnection at the dayside magnetopause.
Ahead of fast magnetic clouds, interplanetary shocks are often
observed
[Burlaga, 1995].

Figure 2

Normally, magnetic clouds have low Alfvén,
MA, and
magnetosonic,
Mms, Mach numbers
[Farrugia et al.,1995].
Thus
the Earth's bow shock may be expected to be displaced sunward with
respect to its statistical position, as given e.g., by
Fairfield [1971].
During the cloud event, the Geotail spacecraft crossed the
magnetosheath on a dawn-to-dusk orbit. Its trajectory is shown
in Figure 2,
where we have superposed on the ( YZ ), ( XZ ), and
( XY ) projections (in GSE coordinates) of the 26 bow shock
crossings, which are all located on dawnside ( Y<0 ) and indicated
by crosses.

Wind proton and magnetic field data are plotted in Figure 3. The
panels
show from top to bottom
the density (cm
-3 ), bulk speed (km s
-1 ), temperature (K),
the GSE
X,Y,Z components of the interplanetary magnetic field (nT),
and
its strength (nT). The bottom two panels show the magnetosonic Mach number
and the solar wind dynamic pressure (nPa).

The magnetic cloud arrived at Wind at approximately 1900 UT on 18 October
1995,
preceded by an interplanetary shock at ~1040 UT.
The magnetic field turned abruptly and strongly
southward when Wind entered the magnetic cloud, and it rotated
gradually to a northward orientation during the next ~24 hours. The
magnetic
field strength in the cloud was large (20-30 nT) and
relatively constant.
Note the relatively constant bulk speed in the cloud.
The magnetosonic Mach number in the cloud is very low (between 2
and 4), which is ideal to check the position of the bow shock
because this is precisely the range where in MHD theories the
standoff distance starts to increase.

Solar wind dynamic pressure is high in the cloud's sheath and very low
inside
the cloud with a gradual increase from ~1 nPa up to ~10 nPa.
This increase is mainly due to the interaction with a
faster trailing stream
[Farrugia et al.,1998].

Most of the time
pdyn is below the historical average of 2.2 nPa.
The interplanetary parameters provide an
ideal situation to examine the bow shock position
as a function of low magnetosonic Mach
number and under a wide range of dynamic pressure from
0.2

dyn<10 nPa
in the cloud.

Geotail Observations

Figure 4

Figure 4 shows an overplot of Wind and Geotail data,
where the Wind data have been shifted by the average delay time
of ~43 min. From top to bottom the figure shows the
solar wind density (cm
-3 ), the solar wind bulk speed
(km s-1 ), the GSE
X,Y,Z components of the interplanetary magnetic
field (nT), and its strength (nT).
Geotail is initially in the solar wind when the sheath of the
cloud passes. When the cloud arrives,
Bz measured by Geotail (GT) suddenly
drops to a large negative value of about
(-45) nT and the
bow shock moves outward and GT is located in the magnetosheath.
Each time GT is in the solar wind we can see good
agreement at the two spacecrafts, and vice versa when the bow shock
moves out and Geotail is in the Earth's magnetosheath. From the
measurements made by Geotail we can identify five different
periods of IMF and plasma behavior of the solar wind (Table 1), three
periods with
bow shock crossings, two without.

Shape and Location of the Bow Shock

In a statistical analysis,
Farris et al. [1991]
studied 351
independent bow shock
crossings and 233 independent magnetopause crossings made by the ISEE 1
spacecraft
from 1977 to 1980 to determine the average positions and shapes of the bow shock
and the magnetopause. They represented the bow shock as a
paraboloid and obtained statistically
X=as-bs(Y2+Z2)
and
as=13.7 0.2RE
and
bs=0.0223 0.0003RE-1
for the subsolar standoff distance and the shape parameters,
respectively.

Specifically for low Alfvén Mach numbers,
Farrugia et al. [1995]
derived a
quasi-linear
connection between the thickness of the magnetosheath
Dms normalized
to the subsolar radius of the magnetopause
amp ( Dms (as-amp)/a
mp ) and the inverse square of the Alfvén
Mach
number,
1/MA2, as it is in our study.

Therefore, ignoring the motion of the bow shock, we fit the
crossings to the
Farris et al. [1991]
formula to a functional form which
brings
out the
1/MA2 dependence explicitly. Instead of two parameters,
as and
bs,
in the Farris formula, a four-parameter formula is employed:

Figure 5 shows the trajectory of Geotail (dotted) in the
XY and
XZ plane approaching the subsolar line from the dawnside.
Crosses on this trajectory mark the 26 bow shock crossing as
seen by Geotail. The solid curve represents the
Farris et al. [1991]
bow shock, whereas the dashed curve shows our fitted bow shock.
With respect to the
Farris et al.
formula, our bow shock is, on average, displaced by 1.85
RE sunward.
If we consider the second period (Table 1) with its
unusually low and rather constant values of
Mms and
pdyn and no bow shock crossings, we may
conclude
that the bow shock must have been sunward of the orbit of Geotail.
Otherwise, crossings occur when either
pdyn and/or
Mms vary; see, for example, period 1 from
1900-2300 UT in Figure 3 in the bottom two panels.

Bow Shock Normals

We employ two different methods of calculating the bow shock
normals:
(1) from the shape of the Farris et al.
bow shock and
(2) from the coplanarity theorem
[after
Abraham-Shrauner and Yun, 1976].

For method (1) we know the position vector
r of the boundary

(2)

Thus the shock normal vector at any point at the curve can be derived
from
vector analysis

(3)

For the shock normal derived from the coplanarity theorem we compute
upstream and downstream values of the magnetic field and obtain

(4)

where subscripts 1 and 2 refer to upstream and downstream values
of
B.

Figure 6

Figure 7

Figure 6 shows the Farris et al.
bow shock shape and the normals,
mentioned above, averaged for the three periods of shock
crossings. The solid line normal refers to
calculation 1
and the
dotted line to
calculation 2, respectively.

In Figure 7
we plot for each interval
the angle
l between the derived shock
normals and the subsolar line, also for each method.
The observed normal directions have large scatter, which however
decreases
in groups 2 and 3, i.e., as Geotail approaches the subsolar line. The large
scatter of the coplanarity normals in group 1 (at the flanks of
the bow shock) may be due to localized disturbances on the shock
and hint to a more fluttery shock shape at the flanks.
The last group, where the scatter is small still has
Dl= 4.6o.
This may indicate that the actual bow
shock shape departs from an axisymmetrical shape, what may be due
to the large
By component of the cloud field at this time.

The angles
q between the shock normals and the
IMF
Bn at each bow shock
crossing are all
q>45o, and thus all shock crossings
are perpendicular shocks.

Velocity of the Bow Shock

Now we use the coplanarity normals to derive the bow shock
velocity after
Burgess [1995]

(5)

Figure 8

The velocities of the bow shock at each crossing are plotted
in
Figure 8.
Crosses and triangles mark whether the bow shock is moving
outward or inward. The first and the last group of crossings all have a
velocity of the order of ~250 km s-1, whereas the second
group has a large scatter and lower velocities down to
~20 km s-1, what is probably due to the small density jumps across
the bow shock during period 2 (see Geotail measurements in Figure 4).

Effects of Dynamic Pressure and Mach Numbers

The magnetosonic Mach number is very low at
times of shock in and out motions,
between 1.2 and 3.
The trend for large sunward displacement for decreasing
Mms is evident
here.
It has been shown in previous studies
[e.g.,
Cairns and Grabbe, 1994;
Cairns and Lyon, 1995, 1996;
Cairns et al.,1995;
Fairfield, 1971;
Farris et al.,1991;
Formisano et al., 1971;
Grabbe,1997;
Peredo et al.,1995]
that at very low Alfvén
and magnetosonic Mach numbers the subsolar distance
could increase up to 30 or more
RE.
Note, however,
that we never observe a static bow shock but one moving
either earthward or sunward.

We now discuss the dynamic pressure.
For an increasing dynamic pressure, the magnetopause standoff distance moves
inward, as does the bow shock. We assume here that
this is the primary effect of dynamic pressure. We shall therefore not study
changes of the shape of the magnetosphere (blunt to more pointed), which rapid
and large dynamic pressure changes may be expected to occasion; that is,
we shall consider in first approximation only changes in dynamic pressure,
which are slow, i.e., which affect the whole magnetosphere.
The crossings are obviously correlated with changes in dynamic pressure.
When the dynamic pressure is low and
<1 nPa,
there are no crossings at all;
that is, the
Mms and the
pdyn effects on the bow shock position
act in the same direction.

Figure 9

In Figure 9 we superpose dynamic pressure, magnetosonic Mach number,
and
the magnetic field at Geotail
for the three sets of crossings. The figure shows
that there is a clear extra response delay of about 10-20 min,
for both inward and outward motion (see, e.g., rise of
pdyn at 2105
UT in the top panel and
at 3225 UT
in the middle panel). This is probably mainly due to
the delay for changes in
Mms and
pdyn seen at Geotail to
reach the bow shock and subsequently for the bow shock to cross the
Geotail position.
For outward motions it could be that
pdyn and
Mms change
slowly, and the bow shock approached Geotail without crossing it,
but it does
later after a further impulsive drop in magnetosonic Mach number.

Much work has been done on
the bow shock standoff distance as a function of interplanetary
parameters
[see e.g.
Grabbe and Cairns, 1995, and references therein].
In recent years there is renewed interest on this issue for cases when the
Alfvén Mach number is low
[Cairns and Grabbe, 1994;
Cairns et al., 1995;
Russell and Petrinec, 1996a, 1996b].
In their paper,
Grabbe and Cairns [1995]
present an analytical MHD formula for
the density jump
r2/r1=X

Because of the perturbation technique used to derive this formula,
it is valid only for values of
q60o.
In our
case, where the average value of
q75o,
one has to
take a simplified solution also presented by
Grabbe and Cairns [1995]

(6)

with

(7)

An empirical relation between the bow shock standoff distance
( as ), the magnetopause nose ( amp ), and
X takes the following
form
[Cairns and Lyon,1995;
Farris and Russell,1994;
Seiff, 1962;
Spreiter et al.,1966]

(8)

For the gas dynamic empirical relation found by
Seiff [1962]
and further
developed by
Spreiter et al. [1966],
j=1 and
k=1.1, where the value of
k depends on the obstacle shape. In the model presented by
Farris and Russell [1994]
the value for
k is modified at lower Mach
numbers by
k=1.1Mms2/(Mms2-1),
while
j stays at 1. In
the model developed from MHD simulations by
Cairns and Lyon [1995],
j=0.4 and
k=3.4 for quasi-perpendicular flows with
MS8 and
MA>1.5. These values are appropriate for our
problem, and so we calculate the ratio

(9)

using (6) and (7) for
X.

Figure 10

Figure 11

Figure 10 shows
as/amp during the passage of the cloud.
The
greatest value can be seen at about 2345 UT when Alfvén
Mach
number and dynamic pressure reach their lowest values. Compared
with the predictions of
Cairns and Lyon [1995, Figure 3]
our results
qualitatively agree fairly well in the studied range of
MA and
Mms, respectively.

Figure 11 shows six panels where the first one contains the
predicted
as from (9) (solid line) and the given position dependent
on
MA,
keeping the dynamic pressure at its average value for the first group of
crossings.
In the second panel we keep the Mach number at its average value for the group of
crossings and check the effects of
pdyn through parameter
amp in
(9). The other four panels repeat this procedure for the other two
groups of crossings. The most impressive thing which can be made
out of this figure is that it seems that especially for the large
upstream
excursions of the bow shock,
MA influences the bow
shock motion most. Of course, when looking at the solar wind
parameters, this is an unexpected result, because of fairly
constant values of
B and the proportionality of
pdyn and
MA via the solar wind density and bulk speed
( MA2=m0rvsw2/B2=m0pdyn/B2 ).
Two considerations have to be taken into account when analyzing this figure:
(1)
as is calculated in subsolar distance, and our crossings
are
not subsolar; (2) to derive the nose of the magnetopause, we have
used the formula for pressure balance, which might not give the
most realistic behavior of the magnetopause for this event.
The very large, negative
Bz

Figure 12

(see Figure 12)
should lead to
magnetic field line reconnection and to an erosion of the
magnetosphere. Thus the magnetopause calculated from pressure
balance should be an overestimation of the true standoff distance.

Figure 13

For Figure 13 we plotted four different ratios
as/amp for the
26 measured bow shock crossings. The first one
repeats the Cairns and Lyon formula (9),
which takes into account the
plasma and magnetic field data measured by Geotail. For
the other three calculations the bow shock standoff
distance is taken from the Geotail crossings brought to the
subsolar line via the fitted bow shock shape described in (1).
For the magnetopause standoff distance we use various calculations,
which also underlay some restrictions, because
of our set of solar wind data.
In this way we combine actual measurements
with theory and models, respectively.
(1) Shue et al. [1998] (dotted line):
This is a numerical
formula that takes into account the possibility of erosion but is
also restricted in the range of negative
Bz -18 nT;
pressure balance (dashed line):
This simple formula ignores
Bz;
(2) Farrugia et al. [1995] (dashed dotted line):
The magnetopause is taken as a tangential discontinuity, which
precludes reconnection. On the other hand, the relation
was derived specifically for low Alfvén
Mach number.
The dependence of
the magnetopause thickness is normalized to
amp of
1/MA2,
which is an important feature in our study.

From the figure we can see that for the first period, where we have
very negative
Bz, the Cairns and Lyon formula and the Shue et
al. formula fit quite well; and in the third period with positive
Bz, the Farrugia et al. magnetopause leads to rather good
agreement with Grabbe and Cairns.

Conclusions

1. We examined 26 repeated crossings of the bow shock on
18-19 October 1995, made by Geotail.

2. The period studied corresponded to an Earth passage of an
interplanetary magnetic cloud.

3. We related these crossings to interplanetary parameters,
the solar wind dynamic pressure, and the solar wind Alfvén
and
magnetosonic Mach numbers. For the interval studied,
the ranges of these parameters were
1ms<4 and
0.2nPa

dyn<10nPa,
respectively. Thus
we expect large sunward displacements of the bow shock.

4. Compared to the model bow shock of Farris et al., we find a net
average sunward displacement of 1.85
RE due to the low Alfvén
Mach number.

5. We calculated the bow shock normals in two different ways and found
that the
coplanarity normals agree with the Farris et al. shape normals except near
the flanks, where
a wide scatter in the derived normals is observed.

8. We examine a delay in the response time
of the bow shock between
Mms and
Pdyn changes
at Geotail and the bow shock crossings. This delay was of the order of
~10-20 min.

9. Our results are in fair agreement with the simulations of Cairns and
Lyon on the standoff
bow shock position in relation to
Mms.

10. We compare the position of the magnetopause and bow shock as predicted by
various models and
offered reasons for discrepancies between them.

11. The drawing
of any conclusions due to the extreme conditions of the interplanetary
magnetic field should also have been part of bow shock observations in this
special magnetic cloud event.
As seen from the data plots, there was strong negative
Bzfor a long period then rotating
to the northward direction,
also a strong eastward
component rotating to strong westward values.
However, reconnection might occur, the
magnetopause could be eroded, and
asymmetries in the Earth magnetosphere could
play a nonnegligible role.
This we point out in Figure 13,
comparing the
as/amp values in
the first panel when
Bz was less than zero. In this panel the
result of the Cairns and Lyon model agrees rather well with the
Shue et al. formula, which takes into account the direction of
Bz. The pressure balance results show much lower values.
Vice
versa in the third panel with
Bz>0, the
as/amp derived from
the actual SW characteristics fit better with pressure balance
than with Shue et al.
Further work will be reported elsewhere
[Farrugia et al., 2001].

Acknowledgments

This work is partially supported by the INTAS-ESA project 99-01277,
the Austrian "Fonds zur Förderung der wissenschaftlichen
Forschung" under projects P13804-TPH and P12761-TPH, by NASA grant NAG5-2834, by
grant 98-05-65290 from the Russian Foundation of Basic Research, by grant 97-0-13.0-71
from the Russian Ministry of Education, and by the Austrian Academy of Sciences,
"Verwaltungstelle für Auslandsbeziehungen."